Question: All of the 4th grade teachers and students from Loyola went on a field trip to an art museum. Tickets were $$7.50$ each for teachers and $$3.50$ each for students, and the group paid $$40.00$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$15.00$ each for teachers and $$8.00$ each for students, and the group paid $$85.00$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7.5x+3.5y = 40}$ ${15x+8y = 85}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-15x-7y = -80}$ ${15x+8y = 85}$ Add the top and bottom equations together. ${y = 5}$ Now that you know ${y = 5}$ , plug it back into $ {7.5x+3.5y = 40}$ to find $x$ ${7.5x + 3.5}{(5)}{= 40}$ $7.5x+17.5 = 40$ $7.5x = 22.5$ $x = \dfrac{22.5}{7.5}$ ${x = 3}$ You can also plug ${y = 5}$ into $ {15x+8y = 85}$ and get the same answer for $x$ ${15x + 8}{(5)}{= 85}$ ${x = 3}$ There were $3$ teachers and $5$ students on the field trips.